In order to model contribution of specular components to the MIMO channel transfer function we consider first a contribution from a single specular component.. The angular spread on both
Trang 1where φ n is a randomly distributed phase with the variance given by equation (18) If σ φ2
4π2the model reduces to well accepted spherically symmetric diffusion component model; if
σ φ2=0, LoS-like conditions for specular component are observed with the rest of the values
spanning an intermediate scenario
Detailed investigation of statistical properties of the model, given by equation (20), can be
found in (Beckmann and Spizzichino; 1963) and some consequent publications, especially in
the field of optics (Barakat; 1986), (Jakeman and Tough; 1987) Assuming that the Central
Limit Theorem holds, as in (Beckmann and Spizzichino; 1963), one comes to conclusion that
ξ=ξ I+jξ Q is a Gaussian process with zero mean and unequal variances σ2
I and σ2
Qof the real
and imaginary parts Therefore ξ is an improper random process (Schreier and Scharf; 2003).
Coupled with a constant term m=m I+jm Qfrom the LoS type components, the model (20)
gives rise to a large number of different distributions of the channel magnitude, including
Rayleigh (m=0, σ I=σ Q ), Rice (m = 0, σ I =σ Q ), Hoyt (m = 0, σ I > 0 σ Q=0) and many
others (Klovski; 1982), (Simon and Alouini; 2000) Following (Klovski; 1982) we will refer to
the general case as a four-parametric distribution, defined by the following parameters
m2
I+m2
Q , φ=arctanm Q
q2 = m2
I+m2
Q
σ2I +σ Q2 , β=σ
2
Q
Two parameters, q2and β, are the most fundamental since they describe power ration between
the deterministic and stochastic components (q2) and asymmetry of the components (β) The
further study is focused on these two parameters
2.2.2 Channel matrix model
Let us consider a MIMO channel which is formed by N T transmit and N Rreceived antennas
The N R × N Tchannel matrix
can be decomposed into three components Line of sight component HLoScould be
repre-sented as
HLoS=
P LoS
N T N RaLbH
Here P LoSis power carried by LoS component, aLand bL are receive and transmit antenna
manifolds (van Trees; 2002) and φ LoSis a deterministic constant phase Elements of both
man-ifold vectors have unity amplitudes and describe phase shifts in each antenna with respect to
some reference point1 Elements of the matrix Hdi f f are assumed to be drawn from proper
(spherically-symmetric) complex Gaussian random variables with zero mean and correlation
between its elements, imposed by the joint distribution of angles of arrival and departure
(Almers et al.; 2006) This is due to the assumption that the diffusion component is composed
of a large number of waves with independent and uniformly distributed phases due to large
and rough scattering surfaces Both LoS and diffusive components are well studied in the
literature Combination of the two lead to well known Rice model of MIMO channels (Almers
et al.; 2006)
1 This is not true when the elements of the antenna arrays are not identical or different polarizations are
used.
Proper statistical interpretation of specular component Hspis much less developed in MIMO literature, despite its applications in optics and random surface scattering (Beckmann and Spizzichino; 1963) The specular components represent an intermediate case between LoS and
a purely diffusive component Formation of such a component is often caused by mild rough-ness, therefore the phases of different partial waves have either strongly correlated phases or non-uniform phases
In order to model contribution of specular components to the MIMO channel transfer function
we consider first a contribution from a single specular component Such a contribution could
be easily written in the following form
Hsp=
P sp
N T N R[a wa] [b wb]H ξ (25)
Here P sp is power of the specular component, ξ=ξ R+jξ Iis a random variable drawn
accord-ing to equation (20) from a complex Gaussian distribution with parameters m I+jm Q , σ2
I , σ2
Q
and independent in-phase and quadrature components Since specular reflection from a mod-erately rough or very rough surface allows reflected waves to be radiated from the first Fresnel zone it appears as a signal with some angular spread This is reflected by the window terms
waand wb(van Trees; 2002; Primak and Sejdi´c; 2008) It is shown in (Primak and Sejdi´c; 2008) that it could be well approximated by so called discrete prolate spheroidal sequences (DPSS) (Percival and Walden; 1993b) or by a Kaiser window (van Trees; 2002; Percival and Walden; 1993b) If there are multiple specular components, formed by different reflective rough sur-faces, such as in an urban canyon in Fig 1, the resulting specular component is a weighted sum of (25) like terms defined for different angles of arrival and departures:
Hsp=∑
k=1
P sp,k
N T N R
ak wa,k bk wb,k H ξ k (26)
It is important to mention that in the mixture (26), unlike the LoS component, the absolute
value of the mean term is not the same for different elements of the matrix Hsp Therefore, it
is not possible to model them as identically distributed random variables Their parameters (mean values) also have to be estimated individually However, if the angular spread of each
specular component is very narrow, the windows wa,kand wb,k could be assumed to have only unity elements In this case, variances of the in-phase and quadrature components of all
elements of matrix Hspare the same
3 MDPSS wideband simulator of Mobile-to-Mobile Channel
There are different ways of describing statistical properties of wide-band time-variant MIMO channels and their simulation The most generic and abstract way is to utilize the time varying
impulse response H(τ, t)or the time-varying transfer function H(ω, t)(Jeruchim et al.; 2000), (Almers et al.; 2006) Such description does not require detailed knowledge of the actual channel geometry and is often available from measurements It also could be directly used in simulations (Jeruchim et al.; 2000) However, it does not provide good insight into the effects
of the channel geometry on characteristics such as channel capacity, predictability, etc In
addition such representations combine propagation environment with antenna characteristics into a single object
Trang 2An alternative approach, based on describing the propagation environment as a collection
of scattering clusters is advocated in a number of recent publications and standards (Almers
et al.; 2006; Asplund et al.; 2006) Such an approach gives rise to a family of so called
Sum-Of-Sinusoids (SoS) simulators
Sum of Sinusoids (SoS) or Sum of Cisoids (SoC) simulators (Patzold; 2002; SCM Editors; 2006)
is a popular way of building channel simulators both in SISO and MIMO cases However,
this approach is not a very good option when prediction is considered since it represents a
signal as a sum of coherent components with large prediction horizon (Papoulis; 1991) In
addition it is recommended that up to 10 sinusoids are used per cluster In this
communi-cation we develop a novel approach which allows one to avoid this difficulty The idea of a
simulator combines representation of the scattering environment advocated in (SCM Editors;
2006; Almers et al.; 2006; Molisch et al.; 2006; Asplund et al.; 2006; Molish; 2004) and the
ap-proach for a single cluster environment used in (Fechtel; 1993; Alcocer et al.; 2005) with some
important modifications (Yip and Ng; 1997; Xiao et al.; 2005)
3.1 Single Cluster Simulator
3.1.1 Geometry of the problem
Let us first consider a single cluster scattering environment, shown in Fig 2 It is assumed
that both sides of the link are equipped with multielement linear array antennas and both
are mobile The transmit array has N T isotropic elements separated by distance d Twhile the
receive side has N R antennas separated by distance d R Both antennas are assumed to be in
the horizontal plane; however extension on the general case is straightforward The antennas
are moving with velocities v T and v Rrespectively such that the angle between corresponding
broadside vectors and the velocity vectors are α T and α R Furthermore, it is assumed that the
impulse response H(τ, t)is sampled at the rate F st , i.e τ=n/F stand the channel is sounded
with the rate F s impulse responses per second, i.e t=m/F s The carrier frequency is f0
Practical values will be given in Section 4
The space between the antennas consist of a single scattering cluster whose center is seen
at the the azimuth φ 0T and co-elevation θ T from the receiver side and the azimuth φ 0Rand
co-elevation θ R The angular spread in the azimuthal plane is ∆φ T on the receiver side and
∆φ Ron the transmit side No spread is assumed in the co-elevation dimension to simplify
calculations due to a low array sensitivity to the co-elevation spread We also assume that θ R=
θ T=π/2 to shorten equations Corresponding corrections are rather trivial and are omitted
here to save space The angular spread on both sides is assumed to be small comparing to the
angular resolution of the arrays due to a large distance between the antennas and the scatterer
(van Trees; 2002):
∆φ T (NT 2πλ −1)dT , ∆φ R 2πλ
The cluster also assumed to produce certain delay spread variation, ∆τ, of the impulse
re-sponse due to its finite dimension This spread is assumed to be relatively small, not exceeding
a few sampling intervals T s=1/F st
3.1.2 Statistical description
It is well known that the angular spread (dispersion) in the impulse response leads to spatial
selectivity (Fleury; 2000) which could be described by corresponding covariance function
ρ(d) = π
−πexp
j2π d
λ φ
Fig 3 Geometry of a single cluster problem
where p(φ)is the distribution of the AoA or AoD Since the angular size of clusters is assumed
to be much smaller that the antenna angular resolution, one can further assume the follow-ing simplifications: a) the distribution of AoA/AoD is uniform and b) the joint distribution
p2(φT , φ R)of AoA/AoD is given by
p2(φT , φ R) =p φ T(φT)pφ R(φR) = 1
∆φ T
·∆1φ R
(29)
It was shown in (Salz and Winters; 1994) that corresponding spatial covariance functions are modulated sinc functions
ρ(d) ≈exp j 2πd
λ sin φ0 sinc ∆φ d
The correlation function of the form (30) gives rise to a correlation matrix between antenna ele-ments which can be decomposed in terms of frequency modulated Discrete Prolate Spheroidal Sequences (MDPSS) (Alcocer et al.; 2005; Slepian; 1978; Sejdi´c et al.; 2008):
R≈WUΛUHWH=
D
∑
k=0
λ kukuH
where Λ≈ID is the diagonal matrix of size D × D (Slepian; 1978), U is N × D matrix of the
dis-crete prolate spheroidal sequences and W=diag{exp(j2πd/λsinndA)} Here d Ais distance
between the antenna elements, N number of antennas, 1 ≤ n ≤ N and D ≈ 2∆φ d λ cos φ0 +1
is the effective number of degrees of freedom generated by the process with the given
covari-ance matrix R For narrow spread clusters the number of degrees of freedom is much less
than the number of antennas D N (Slepian; 1978) Thus, it could be inferred from
equa-tion (31) that the desired channel impulse response H(ω, τ)could be represented as a double sum(tensor product)
H(ω, t) =
D T
∑
n t
D R
∑
n r
λ n t λ n ru(r)n r u(t)n t H h n t ,n r(ω, t) (32)
In the extreme case of a very narrow angular spread on both sides, D R=D T=1 and u(r)1 and
u1(t)are well approximated by the Kaiser windows (Thomson; 1982) The channel correspond-ing to a scorrespond-ingle scatterer is of course a rank one channel given by
H(ω, t) =u(r)1 u(t)1 H h(ω , t) (33)
Trang 3An alternative approach, based on describing the propagation environment as a collection
of scattering clusters is advocated in a number of recent publications and standards (Almers
et al.; 2006; Asplund et al.; 2006) Such an approach gives rise to a family of so called
Sum-Of-Sinusoids (SoS) simulators
Sum of Sinusoids (SoS) or Sum of Cisoids (SoC) simulators (Patzold; 2002; SCM Editors; 2006)
is a popular way of building channel simulators both in SISO and MIMO cases However,
this approach is not a very good option when prediction is considered since it represents a
signal as a sum of coherent components with large prediction horizon (Papoulis; 1991) In
addition it is recommended that up to 10 sinusoids are used per cluster In this
communi-cation we develop a novel approach which allows one to avoid this difficulty The idea of a
simulator combines representation of the scattering environment advocated in (SCM Editors;
2006; Almers et al.; 2006; Molisch et al.; 2006; Asplund et al.; 2006; Molish; 2004) and the
ap-proach for a single cluster environment used in (Fechtel; 1993; Alcocer et al.; 2005) with some
important modifications (Yip and Ng; 1997; Xiao et al.; 2005)
3.1 Single Cluster Simulator
3.1.1 Geometry of the problem
Let us first consider a single cluster scattering environment, shown in Fig 2 It is assumed
that both sides of the link are equipped with multielement linear array antennas and both
are mobile The transmit array has N T isotropic elements separated by distance d Twhile the
receive side has N R antennas separated by distance d R Both antennas are assumed to be in
the horizontal plane; however extension on the general case is straightforward The antennas
are moving with velocities v T and v Rrespectively such that the angle between corresponding
broadside vectors and the velocity vectors are α T and α R Furthermore, it is assumed that the
impulse response H(τ, t)is sampled at the rate F st , i.e τ=n/F stand the channel is sounded
with the rate F s impulse responses per second, i.e t=m/F s The carrier frequency is f0
Practical values will be given in Section 4
The space between the antennas consist of a single scattering cluster whose center is seen
at the the azimuth φ 0T and co-elevation θ T from the receiver side and the azimuth φ 0Rand
co-elevation θ R The angular spread in the azimuthal plane is ∆φ T on the receiver side and
∆φ Ron the transmit side No spread is assumed in the co-elevation dimension to simplify
calculations due to a low array sensitivity to the co-elevation spread We also assume that θ R=
θ T=π/2 to shorten equations Corresponding corrections are rather trivial and are omitted
here to save space The angular spread on both sides is assumed to be small comparing to the
angular resolution of the arrays due to a large distance between the antennas and the scatterer
(van Trees; 2002):
∆φ T (NT 2πλ −1)dT , ∆φ R 2πλ
The cluster also assumed to produce certain delay spread variation, ∆τ, of the impulse
re-sponse due to its finite dimension This spread is assumed to be relatively small, not exceeding
a few sampling intervals T s=1/F st
3.1.2 Statistical description
It is well known that the angular spread (dispersion) in the impulse response leads to spatial
selectivity (Fleury; 2000) which could be described by corresponding covariance function
ρ(d) = π
−πexp
j2π d
λ φ
Fig 3 Geometry of a single cluster problem
where p(φ)is the distribution of the AoA or AoD Since the angular size of clusters is assumed
to be much smaller that the antenna angular resolution, one can further assume the follow-ing simplifications: a) the distribution of AoA/AoD is uniform and b) the joint distribution
p2(φT , φ R)of AoA/AoD is given by
p2(φT , φ R) =p φ T(φT)pφ R(φR) = 1
∆φ T
·∆1φ R
(29)
It was shown in (Salz and Winters; 1994) that corresponding spatial covariance functions are modulated sinc functions
ρ(d) ≈exp j 2πd
λ sin φ0 sinc ∆φ d
The correlation function of the form (30) gives rise to a correlation matrix between antenna ele-ments which can be decomposed in terms of frequency modulated Discrete Prolate Spheroidal Sequences (MDPSS) (Alcocer et al.; 2005; Slepian; 1978; Sejdi´c et al.; 2008):
R≈WUΛUHWH=
D
∑
k=0
λ kukuH
where Λ≈ID is the diagonal matrix of size D × D (Slepian; 1978), U is N × D matrix of the
dis-crete prolate spheroidal sequences and W=diag{exp(j2πd/λsinndA)} Here d Ais distance
between the antenna elements, N number of antennas, 1 ≤ n ≤ N and D ≈ 2∆φ d λ cos φ0 +1
is the effective number of degrees of freedom generated by the process with the given
covari-ance matrix R For narrow spread clusters the number of degrees of freedom is much less
than the number of antennas D N (Slepian; 1978) Thus, it could be inferred from
equa-tion (31) that the desired channel impulse response H(ω, τ)could be represented as a double sum(tensor product)
H(ω, t) =
D T
∑
n t
D R
∑
n r
λ n t λ n ru(r)n ru(t)n t H h n t ,n r(ω, t) (32)
In the extreme case of a very narrow angular spread on both sides, D R=D T=1 and u(r)1 and
u(t)1 are well approximated by the Kaiser windows (Thomson; 1982) The channel correspond-ing to a scorrespond-ingle scatterer is of course a rank one channel given by
H(ω, t) =u(r)1 u(t)1 H h(ω , t) (33)
Trang 4Considering the shape of the functions u(r)1 and u1(t) one can conclude that in this scenario
angular spread is achieved by modulating the amplitude of the spatial response of the channel
on both sides It is also worth noting that representation (32) is the Karhunen-Loeve series
(van Trees; 2001) in spatial domain and therefore produces smallest number of terms needed
to represent the process selectivity in spatial domain It is also easy to see that such modulation
becomes important only when the number of antennas is significant
Similar results could be obtained in frequency and Doppler domains Let us assume that
τ is the mean delay associated with the cluster and ∆τ is corresponding delay spread In
addition let it be desired to provide a proper representation of the process in the bandwidth
[− W : W] using N Fequally spaced samples Assuming that the variation of power is relatively
minor within ∆τ delay window, we once again recognize that the variation of the channel
in frequency domain can be described as a sum of modulated DPSS of length N Fand the
time bandwidth product W∆τ The number of MDPSS needed for such representation is
approximately D F=2W∆τ+1 (Slepian; 1978):
h(ω, t) =
D f
∑
n f=1
Finally, in the Doppler domain, the mean resulting Doppler spread could be calculated as
f D= f0
c [vTcos(φT0 − α T) +v Rcos(φR0 − α R)]. (35)
The angular extent of the cluster from sides causes the Doppler spectrum to widen by the
folowing
∆ f D= f0
c [vT ∆φ T v T |sin(φT0 − α T)| + v R ∆φ R |sin(φR0 − α R)|]. (36)
Once again, due to a small angular extent of the cluster it could be assumed that the widening
of the Doppler spectrum is relatively narrow and no variation within the Doppler spectrum
is of importance Therefore, if it is desired to simulate the channel on the interval of time
[0 : T max]then this could be accomplished by adding D=2∆ f D Tmax+1 MDPS:
hd=
D
∑
n d=0
ξ n d
where ξ n dare independent zero mean complex Gaussian random variables of unit variance
Finally, the derived representation could be summarized in tensor notation as follows Let
u(t)n t, u(r)n r, u(ω)
n f and u(d)n d be DPSS corresponding to the transmit, receive, frequency and
Doppler dimensions of the signal with the “domain-dual domain” products (Slepian; 1978)
given by ∆φ T d
λ cos φ T0 , ∆φ R d
λ cos φ R0 , W∆τ and T max ∆ f D respectively Then a sample of a MIMO frequency selective channel with corresponding characteristics could be generated as
D T
∑
n t
D R
∑
n r
D F
∑
n f
d
∑
n d
λ(t)n t λ(r)n r λ (ω) n f λ(T)n d ξ n t ,n r ,n f ,n d ·
1u(r)n t ×2u(r)n r ×3u(ω) n f ×4u(d)n d (38)
whereW4is a tensor composed of modulating sinusoids
w(r) =
1,exp j2π d R
λ
λ(NR −1)
T
w(t) =
1,exp
j2π d T λ
,· · ·,exp
j2π d T
λ(N T −1)
T
(40)
w(ω) = [1,exp(j2π∆Fτ),· · ·,exp(j2π∆F(NF −1))]T
w(d) = [1,exp(j2π∆ fD T s),· · ·,exp(j2π∆ fD(Tmax − T s))]T
(41) andis the Hadamard (element wise) product of two tensors (van Trees; 2002)
3.2 Multi-Cluster environment
The generalization of the model suggested in Section 3.1 to a real multi-cluster environment
is straightforward The channel between the transmitter and the receiver is represented as a set of clusters, each described as in Section (3.1) The total impulse response is superposition
of independently generated impulse response tensors from each cluster
N c −1
∑
k=0
P k H4(k), ∑N c
k=1
where N cis the total number of clusters,H4(k)is a normalized response from the k-th cluster
||H4(k)||2F=1 and P k ≥ 0 represents relative power of k-th cluster and P is the total power.
It is important to mention here that such a representation does not necessarily correspond to a physical cluster distribution It rather reflects interplay between radiated and received signals, arriving from certain direction with a certain excess delay, ignoring particular mechanism of propagation Therefore it is possible, for example, to have two clusters with the same AoA and AoD but a different excess delay Alternatively, it is possible to have two clusters which correspond to the same AoD and excess delay but very different AoA
Equations (38) and (42) reveal a connection between Sum of Cisoids (SoC) approach (SCM Editors; 2006) and the suggested algorithms: one can consider (38) as a modulated Cisoid Therefore, the simulator suggested above could be considered as a Sum of Modulated Cisoids simulator
In addition to space dispersive components, the channel impulse response may contain a number of highly coherent components, which can be modelled as pure complex exponents Such components described either direct LoS path or specularly reflected rays with very small phase diffusion in time Therefore equation (42) should be modified to account for such com-ponents:
1
1+K
N c −1
∑
k=0
P ck H4(k) +
K
1+K
N s −1
∑
k=0
Here N s is a number of specular components including LoS and K is a generalized Rice factor describing ratio between powers of specular P sk and non-coherent/diffusive components P ck
K= ∑N s −1 k=0 P sk
∑N c −1
Trang 5Considering the shape of the functions u(r)1 and u(t)1 one can conclude that in this scenario
angular spread is achieved by modulating the amplitude of the spatial response of the channel
on both sides It is also worth noting that representation (32) is the Karhunen-Loeve series
(van Trees; 2001) in spatial domain and therefore produces smallest number of terms needed
to represent the process selectivity in spatial domain It is also easy to see that such modulation
becomes important only when the number of antennas is significant
Similar results could be obtained in frequency and Doppler domains Let us assume that
τ is the mean delay associated with the cluster and ∆τ is corresponding delay spread In
addition let it be desired to provide a proper representation of the process in the bandwidth
[− W : W] using N Fequally spaced samples Assuming that the variation of power is relatively
minor within ∆τ delay window, we once again recognize that the variation of the channel
in frequency domain can be described as a sum of modulated DPSS of length N F and the
time bandwidth product W∆τ The number of MDPSS needed for such representation is
approximately D F=2W∆τ+1 (Slepian; 1978):
h(ω, t) =
D f
∑
n f=1
Finally, in the Doppler domain, the mean resulting Doppler spread could be calculated as
f D= f0
c [vTcos(φT0 − α T) +v Rcos(φR0 − α R)]. (35)
The angular extent of the cluster from sides causes the Doppler spectrum to widen by the
folowing
∆ f D= f0
c [vT ∆φ T v T |sin(φT0 − α T)| + v R ∆φ R |sin(φR0 − α R)|]. (36)
Once again, due to a small angular extent of the cluster it could be assumed that the widening
of the Doppler spectrum is relatively narrow and no variation within the Doppler spectrum
is of importance Therefore, if it is desired to simulate the channel on the interval of time
[0 : T max]then this could be accomplished by adding D=2∆ f D Tmax+1 MDPS:
hd=
D
∑
n d=0
ξ n d
where ξ n dare independent zero mean complex Gaussian random variables of unit variance
Finally, the derived representation could be summarized in tensor notation as follows Let
u(t)n t, u(r)n r, u(ω)
n f and u(d)n d be DPSS corresponding to the transmit, receive, frequency and
Doppler dimensions of the signal with the “domain-dual domain” products (Slepian; 1978)
given by ∆φ T d
λ cos φ T0 , ∆φ R d
λ cos φ R0 , W∆τ and T max ∆ f D respectively Then a sample of a MIMO frequency selective channel with corresponding characteristics could be generated as
D T
∑
n t
D R
∑
n r
D F
∑
n f
d
∑
n d
λ(t)n t λ(r)n r λ (ω) n f λ(T)n d ξ n t ,n r ,n f ,n d ·
1u(r)n t ×2u(r)n r ×3u(ω) n f ×4u(d)n d (38)
whereW4is a tensor composed of modulating sinusoids
w(r) =
1,exp j2π d R
λ
λ(N R −1)
T
w(t) =
1,exp
j2π d T λ
,· · ·,exp
j2π d T
λ (NT −1)
T
(40)
w(ω) = [1,exp(j2π∆Fτ),· · ·,exp(j2π∆F(NF −1))]T
w(d) = [1,exp(j2π∆ fD T s),· · ·,exp(j2π∆ fD(Tmax − T s))]T
(41) andis the Hadamard (element wise) product of two tensors (van Trees; 2002)
3.2 Multi-Cluster environment
The generalization of the model suggested in Section 3.1 to a real multi-cluster environment
is straightforward The channel between the transmitter and the receiver is represented as a set of clusters, each described as in Section (3.1) The total impulse response is superposition
of independently generated impulse response tensors from each cluster
N c −1
∑
k=0
P k H4(k), ∑N c
k=1
where N cis the total number of clusters,H4(k)is a normalized response from the k-th cluster
||H4(k)||2F=1 and P k ≥ 0 represents relative power of k-th cluster and P is the total power.
It is important to mention here that such a representation does not necessarily correspond to a physical cluster distribution It rather reflects interplay between radiated and received signals, arriving from certain direction with a certain excess delay, ignoring particular mechanism of propagation Therefore it is possible, for example, to have two clusters with the same AoA and AoD but a different excess delay Alternatively, it is possible to have two clusters which correspond to the same AoD and excess delay but very different AoA
Equations (38) and (42) reveal a connection between Sum of Cisoids (SoC) approach (SCM Editors; 2006) and the suggested algorithms: one can consider (38) as a modulated Cisoid Therefore, the simulator suggested above could be considered as a Sum of Modulated Cisoids simulator
In addition to space dispersive components, the channel impulse response may contain a number of highly coherent components, which can be modelled as pure complex exponents Such components described either direct LoS path or specularly reflected rays with very small phase diffusion in time Therefore equation (42) should be modified to account for such com-ponents:
1
1+K
N c −1
∑
k=0
P ck H4(k) +
K
1+K
N s −1
∑
k=0
Here N s is a number of specular components including LoS and K is a generalized Rice factor describing ratio between powers of specular P sk and non-coherent/diffusive components P ck
K=∑N s −1 k=0 P sk
∑N c −1
Trang 6While distribution of the diffusive component is Gaussian by construction, the distribution of
the specular component may not be Gaussian A more detailed analysis is beyond the scope
of this chapter and will be considered elsewhere We also leave a question of identifying and
distinguishing coherent and non-coherent components to a separate manuscript
4 Examples
Fading channel simulators (Jeruchim et al.; 2000) can be used for different purposes The goal
of the simulation often defines not only suitability of a certain method but also dictates choice
of the parameters One possible goal of simulation is to isolate a particular parameter and
study its effect of the system performance Alternatively, a various techniques are needed
to avoid the problem of using the same model for both simulation and analysis of the same
scenario In this section we provide a few examples which show how suggested algorithm
can be used for different situations
4.1 Two cluster model
The first example we consider here is a two-cluster model shown in Fig 4 This geometry is
Fig 4 Geometry of a single cluster problem
the simplest non-trivial model for frequency selective fading However, it allows one to study
effects of parameters such as angular spread, delay spread, correlation between sites on the
channel parameters and a system performance The results of the simulation are shown in
Figs 5-6 In this examples we choose φ T1=20o , φ T2=20o , φ R1=0o , φ R2=110o , τ1=0.2 µs,
τ2=0.4 µs, ∆τ1=0.2 µs, ∆τ2=0.4 µs.
4.2 Environment specified by joint AoA/AoD/ToA distribution
The most general geometrical model of MIMO channel utilizes joint distribution p(φT , φ R , τ),
0≤ φ T < 2π, 0 ≤ φ R < 2π, τ min ≤ τ ≤ τ max, of AoA, AoD and Time of Arrival (ToA) A few of
such models could be found in the literature (Kaiserd et al.; 2006), (Andersen and Blaustaein;
2003; Molisch et al.; 2006; Asplund et al.; 2006; Blaunstein et al.; 2006; Algans et al.; 2002)
Theoretically, this distribution completely describes statistical properties of the MIMO
chan-nel Since the resolution of the antenna arrays on both sides is finite and a finite bandwidth
of the channel is utilized, the continuous distribution p(φT , φ R , τ)can be discredited to
pro-duce narrow “virtual” clusters centered at[φTk , φ Rk , τ k]and with spread ∆φ Tk , ∆φ Rk and ∆τ k
10 −3
10 −2
10 −1
100
Doppler frequency, Hz
Fig 5 PSD of the two cluster channel response
respectively and the power weight
4π2(τmax − τ min)×
τ k+∆τk/2
τ k −∆τ k/2 dτ φ Tk+∆φTk/2
φ Tk −∆φ Tk/2 dφ T
φ Rk+∆φRk/2
φ Rk −∆φ Rk/2 p(φ T , φ R , τ)dφR (45)
We omit discussions about an optimal partitioning of each domain due to the lack of space Assume that each virtual cluster obtained by such partitioning is appropriate in the frame discussed in Section 3.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Delay, µ s
Fig 6 PDP of the two cluster channel response
As an example, let us consider the following scenario, described in (Blaunstein et al.; 2006)
In this case the effect of the two street canyon propagation results into two distinct angles
Trang 7While distribution of the diffusive component is Gaussian by construction, the distribution of
the specular component may not be Gaussian A more detailed analysis is beyond the scope
of this chapter and will be considered elsewhere We also leave a question of identifying and
distinguishing coherent and non-coherent components to a separate manuscript
4 Examples
Fading channel simulators (Jeruchim et al.; 2000) can be used for different purposes The goal
of the simulation often defines not only suitability of a certain method but also dictates choice
of the parameters One possible goal of simulation is to isolate a particular parameter and
study its effect of the system performance Alternatively, a various techniques are needed
to avoid the problem of using the same model for both simulation and analysis of the same
scenario In this section we provide a few examples which show how suggested algorithm
can be used for different situations
4.1 Two cluster model
The first example we consider here is a two-cluster model shown in Fig 4 This geometry is
Fig 4 Geometry of a single cluster problem
the simplest non-trivial model for frequency selective fading However, it allows one to study
effects of parameters such as angular spread, delay spread, correlation between sites on the
channel parameters and a system performance The results of the simulation are shown in
Figs 5-6 In this examples we choose φ T1=20o , φ T2=20o , φ R1=0o , φ R2=110o , τ1=0.2 µs,
τ2=0.4 µs, ∆τ1=0.2 µs, ∆τ2=0.4 µs.
4.2 Environment specified by joint AoA/AoD/ToA distribution
The most general geometrical model of MIMO channel utilizes joint distribution p(φT , φ R , τ),
0≤ φ T < 2π, 0 ≤ φ R < 2π, τ min ≤ τ ≤ τ max, of AoA, AoD and Time of Arrival (ToA) A few of
such models could be found in the literature (Kaiserd et al.; 2006), (Andersen and Blaustaein;
2003; Molisch et al.; 2006; Asplund et al.; 2006; Blaunstein et al.; 2006; Algans et al.; 2002)
Theoretically, this distribution completely describes statistical properties of the MIMO
chan-nel Since the resolution of the antenna arrays on both sides is finite and a finite bandwidth
of the channel is utilized, the continuous distribution p(φT , φ R , τ)can be discredited to
pro-duce narrow “virtual” clusters centered at[φTk , φ Rk , τ k]and with spread ∆φ Tk , ∆φ Rk and ∆τ k
10 −3
10 −2
10 −1
100
Doppler frequency, Hz
Fig 5 PSD of the two cluster channel response
respectively and the power weight
4π2(τmax − τ min)×
τ k+∆τk/2
τ k −∆τ k/2 dτ φ Tk+∆φTk/2
φ Tk −∆φ Tk/2 dφ T
φ Rk+∆φRk/2
φ Rk −∆φ Rk/2 p(φ T , φ R , τ)dφR (45)
We omit discussions about an optimal partitioning of each domain due to the lack of space Assume that each virtual cluster obtained by such partitioning is appropriate in the frame discussed in Section 3.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Delay, µ s
Fig 6 PDP of the two cluster channel response
As an example, let us consider the following scenario, described in (Blaunstein et al.; 2006)
In this case the effect of the two street canyon propagation results into two distinct angles
Trang 8of arrival φ R1=20o and φ R2=50o, AoA spreads roughly of ∆1=∆2=5o and exponential
PDP corresponding to each AoA (see Figs 5 and 6 in (Blaunstein et al.; 2006)) In addition,
an almost uniform AoA on the interval[60 : 80o]corresponds to early delays Therefore, a
simplified model of such environment could be presented by
p(φ R , τ) =
P1 1
∆1exp
τ s1
u(τ − τ1)+
P2 1
∆2exp
τ s2
u(τ − τ2) +
P3 1
∆3exp
τ s3
u(τ − τ3) (46)
where u(t)is the unit step function, τ sk , k=1,2,3 describe rate of decay of PDP By inspection
of Figs 5-6 in (Blaunstein et al.; 2006) we choose τ1=τ2=1.2 ns, τ3=1.1 ns and τ s1=τ s2=
τ s3=0.3 ns Similarly, by inspection of the same figures we assume P1=P2=0.4 and P3=0.2
To model exponential PDP with unit power and average duration τ swe represent it with a set
of N ≥ 1 rectangular PDP of equal energy 1/N The k-th virtual cluster then extends on the
interval[τk−1 : τ k]and has magnitude P k=1/N∆τ k where τ0=0
τ k = τ slnN − k
τ N = τ N−1+ 1
Results of numerical simulation are shown in Figs 7 and 8 It can be seen that a good
agree-ment between the desired characteristics is obtained
10 −3
10 −2
10 −1
10 0
Delay, µ s
Fig 7 Simulated power delay profile for the example of Section 4.2
Similarly, the same technique could be applied to the 3GPP (SCM Editors; 2006) and COST
259 (Asplund et al.; 2006) specifications
10 −3
10 −2
10 −1
100
Normalized Doppler frequency, f/fD
Fig 8 Simulated Doppler power spectral density for the example of Section 4.2
5 MDPSS Frames for channel estimation and prediction 5.1 Modulated Discrete Prolate Spheroidal Sequences
If the DPSS are used for channel estimation, then usually accurate and sparse representations are obtained when both the DPSS and the channel under investigation occupy the same fre-quency band (Zemen and Mecklenbr¨auker; 2005) However, problems arise when the channel
is centered around some frequency| ν o | > 0 and the occupied bandwidth is smaller than 2W,
as shown in Fig 9
Fig 9 Comparison of the bandwidth for a DPSS (solid line) and a channel (dashed line): (a) both have a wide bandwidth; (b) both have narrow bandwidth; (c) a DPSS has a wide
bandwidth, while the channel’s bandwidth is narrow and centered around ν o >0; (d) both have narrow bandwidth, but centered at different frequencies
In such situations, a larger number of DPSS is required to approximate the channel with the same accuracy despite the fact that such narrowband channel is more predictable than a wider band channel (Proakis; 2001) In order to find a better basis we consider so-called Modulated Discrete Prolate Spheroidal Sequences (MDPSS), defined as
M k(N,W,ωm ; n) =exp(jωm n)v k(N,W;n), (50)
Trang 9of arrival φ R1=20o and φ R2=50o, AoA spreads roughly of ∆1=∆2=5oand exponential
PDP corresponding to each AoA (see Figs 5 and 6 in (Blaunstein et al.; 2006)) In addition,
an almost uniform AoA on the interval[60 : 80o]corresponds to early delays Therefore, a
simplified model of such environment could be presented by
p(φ R , τ) =
P1 1
∆1exp
τ s1
u(τ − τ1)+
P2 1
∆2exp
τ s2
u(τ − τ2) +
P3 1
∆3exp
τ s3
u(τ − τ3) (46)
where u(t)is the unit step function, τ sk , k=1,2,3 describe rate of decay of PDP By inspection
of Figs 5-6 in (Blaunstein et al.; 2006) we choose τ1=τ2=1.2 ns, τ3=1.1 ns and τ s1=τ s2=
τ s3=0.3 ns Similarly, by inspection of the same figures we assume P1=P2=0.4 and P3=0.2
To model exponential PDP with unit power and average duration τ swe represent it with a set
of N ≥ 1 rectangular PDP of equal energy 1/N The k-th virtual cluster then extends on the
interval[τk−1 : τ k]and has magnitude P k=1/N∆τ k where τ0=0
τ k = τ slnN − k
τ N = τ N−1+ 1
Results of numerical simulation are shown in Figs 7 and 8 It can be seen that a good
agree-ment between the desired characteristics is obtained
10 −3
10 −2
10 −1
10 0
Delay, µ s
Fig 7 Simulated power delay profile for the example of Section 4.2
Similarly, the same technique could be applied to the 3GPP (SCM Editors; 2006) and COST
259 (Asplund et al.; 2006) specifications
10 −3
10 −2
10 −1
100
Normalized Doppler frequency, f/fD
Fig 8 Simulated Doppler power spectral density for the example of Section 4.2
5 MDPSS Frames for channel estimation and prediction 5.1 Modulated Discrete Prolate Spheroidal Sequences
If the DPSS are used for channel estimation, then usually accurate and sparse representations are obtained when both the DPSS and the channel under investigation occupy the same fre-quency band (Zemen and Mecklenbr¨auker; 2005) However, problems arise when the channel
is centered around some frequency| ν o | > 0 and the occupied bandwidth is smaller than 2W,
as shown in Fig 9
Fig 9 Comparison of the bandwidth for a DPSS (solid line) and a channel (dashed line): (a) both have a wide bandwidth; (b) both have narrow bandwidth; (c) a DPSS has a wide
bandwidth, while the channel’s bandwidth is narrow and centered around ν o >0; (d) both have narrow bandwidth, but centered at different frequencies
In such situations, a larger number of DPSS is required to approximate the channel with the same accuracy despite the fact that such narrowband channel is more predictable than a wider band channel (Proakis; 2001) In order to find a better basis we consider so-called Modulated Discrete Prolate Spheroidal Sequences (MDPSS), defined as
M k(N,W,ωm ; n) =exp(jωm n)v k(N,W;n), (50)
Trang 10where ω m=2πν mis the modulating frequency It is easy to see that MDPSS are also doubly
orthogonal, obey the same equation (7) and are bandlimited to the frequency band[− W+ν:
W+ν]
The next question which needs to be answered is how to properly choose the modulation
frequency ν In the simplest case when the spectrum S(ν)of the channel is confined to a
known band[ν1; ν2], i.e.
S(ν) = 0 ∀ ν ∈ [ ν1, ν2]and| ν1| < | ν2|
the modulating frequency, ν m, and the bandwidth of the DPSS’s are naturally defined by
ν m=ν1+ν2
W=
ν2− ν1
2
as long as both satisfy:
In practical applications the exact frequency band is known only with a certain degree of
accu-racy In addition, especially in mobile applications, the channel is evolving in time Therefore,
only some relatively wide frequency band defined by the velocity of the mobile and the
car-rier frequency is expected to be known In such situations, a one-band-fits-all approach may
not produce a sparse and accurate approximation of the channel To resolve this problem, it
was previously suggested to use a band of bases with different widths to account for different
speeds of the mobile (Zemen et al.; 2005) However, such a representation once again ignores
the fact that the actual channel bandwidth 2W could be much less than 2ν Ddictated by the
maximum normalized Doppler frequency ν D=f D T.
To improve the estimator robustness, we suggest the use of multiple bases, better known as
frames (Kovaˇcevi´c and Chabira; 2007), precomputed in such a way as to reflect various
scat-tering scenarios In order to construct such multiple bases, we assume that a certain estimate
(or rather its upper bound) of the maximum Doppler frequency ν Dis available The first few
bases in the frame are obtained using traditional DPSS with bandwidth 2ν D Additional bases
can be constructed by partitioning the band[− ν D ; ν D]into K subbands with the boundaries of
each subband given by[νk ; ν k+1], where 0≤ k ≤ K − 1, ν k+1 > ν k , and ν0=− ν D , ν K−1=ν D
Hence, each set of MDPSS has a bandwidth equal to ν k+1 − ν kand a modulation frequency
equal to ν m=0.5(νk+ν k+1) Obviously, a set of such functions again forms a basis of functions
limited to the bandwidth[− ν D ; ν D] It is a convention in the signal processing community to
call each basis function an atom While particular partition is arbitrary for every level K ≥1,
we can choose to partition the bandwidth into equal blocks to reduce the amount of stored
precomputed DPSS, or to partition according to the angular resolution of the receive antenna,
etc, as shown in Fig 10.
Representation in the overcomplete basis can be made sparse due to the richness of such a
basis Since the expansion into simple bases is not unique, a fast, convenient and unique
projection algorithm cannot be used Fortunately, efficient algorithms, known generically as
pursuits (Mallat; 1999; Mallat and Zhang; 1993), can be used and they are briefly described in
the next section
Fig 10 Sample partition of the bandwidth for K=4
5.2 Matching Pursuit with MDPSS frames
From the few approaches which can be applied for expansion in overcomplete bases, we choose the so-called matching pursuit (Mallat and Zhang; 1993) The main feature of the algorithm is that when stopped after a few steps, it yields an approximation using only a few atoms (Mallat and Zhang; 1993) The matching pursuit was originally introduced in the sig-nal processing community as an algorithm that decomposes any sigsig-nal into a linear expansion
of waveforms that are selected from a redundant dictionary of functions (Mallat and Zhang; 1993) It is a general, greedy, sparse function approximation scheme based on minimizing the squared error, which iteratively adds new functions (i.e basis functions) to the linear expan-sion In comparison to a basis pursuit, it significantly reduces the computational complexity, since the basis pursuit minimizes a global cost function over all bases present in the dictionary (Mallat and Zhang; 1993) If the dictionary is orthogonal, the method works perfectly Also, to achieve compact representation of the signal, it is necessary that the atoms are representative
of the signal behavior and that the appropriate atoms from the dictionary are chosen
The algorithm for the matching pursuit starts with an initial approximation for the signal, x, and the residual, R:
and it builds up a sequence of sparse approximation stepwise by trying to reduce the norm of
the residue, R= x − x At stage k, it identifies the dictionary atom that best correlates with the
residual and then adds to the current approximation a scalar multiple of that atom, such that
where α k= R(k−1) , φ k / φ k 2 The process continues until the norm of the residual R(k)
does not exceed required margin of error >0:|| R(k)|| ≤ (Mallat and Zhang; 1993) In our
approach, a stopping rule mandates that the number of bases, χ B, needed for signal
approxi-mation should satisfy χ B ≤ 2Nν D +1 Hence, a matching pursuit approximates the signal
using χ Bbases as
x=
χ B
∑
where φ n are χ Bbases from the dictionary with the strongest contributions